Categories & sets in a contradictory world (contd.)

0:00 It was more pressing after the paper by Sayer, as you might have achieved, because he had two occasions. Sayer had to screw the definition of cohomology in his thesis. He explained to Lorel, and then he had to, a major part of his thesis, he had to take all the results of Lorel and he expressed them. But in these people, algebra and geometry are different, because for the first time, Zeiski's topology was natural. But Zeiski's topology came into play with a lot of resistance. There is a story, I don't guarantee, but sometime in 1942 or 43, Zeiski gave a talk about the de-secularization of some races. And in this, in this paper he published a book, and... Look at the collective democracy and at what point is it going to be open? Compactness argument. Compactness argument in exactly the same sense as Erbormuth compactness. Compactness argument in Erbormuth doesn't have to be like Erbormuth. Just say that if you are finally a proof of something, it will be used only finitely in the axiom or finitely in the axiom.

2:30 There, they wanted to get to the end of this area, and Zeiss key topology had to do with compactness. It went on that, in some sense, Zeiss key topology. There are other important things that are now in those places. And so the story was that, to finish this proof, Zeiss key used a compactness algorithm, and then he said, well, now I have to go back to the complex thing. So far, my guess on it was, I'm not going to be looking at this idea, but this time I am. And it seems that Chevalier was in tune with Chevalier's population, and that he said, why don't you just interview everybody that's compact of knowledge, and that he defined, from his point of view, he defined as ideal of knowledge. I suppose it wasn't. Okay, and the idea is that it was Zayaski's topology, but Zayaski's topology was invented to have compactness and it was later developed by Zayaski himself, who said that it's hard to have compactness, but Zayaski's topology is not exactly the one we use, but it's similar to compactness. So, mathematical models are designed to motivate us. And, I think, it's worked by a very good foundation of algebraic geometry. It's so complete. It's the idea of designed geometry that doesn't want to do it.

5:00 So, it divides the body form of how to do algebraic geometry by covering by charts, doing charts, etc. But, it doesn't want to do the topology. It may have been used to optimize the Chevrolet model. Look at the thesis of Chevrolet. The main purpose of the thesis of Chauvelet is to throw away the methods of analytics that were used in classical physics. Of course, as usual, the mathematical method is good, but if you are obsessed by the idea of pure mathematics, I'm not saying I'm very well-acquainted with Chauvelet, and he was always very obsessed by pure, vague mathematics. Used in this moment set by Andre Bey in his 40s, there was some resistance, of course, said to it completely, but I remember that Chevalier was very, very good, and I participated to Chevalier's seminar, possibly done by myself, but then you see that there are many places where he could use this advantage of his eyes, but he doesn't, mostly to note, I mean... Of both, but no measurements. And Borel was, Borel knew very well, he knew it in a fantastic way, and so he knew all the topological arguments, and if you could mimic the topological arguments, you could use a giant number of numerals. But that's not the thing, that's not the thing, that's not the thing, that's not the thing, that's not the thing, that's not the thing, that's not the thing. Going to the top of it. And, so, Otenir asked, I said, okay, obviously, the properties of the categories are such that we can expect about injecting but not projecting it, forget about projecting it, injecting but not projecting it, which is the case of module where particular resources of preservation, of projecting but not projecting, are the chiefs of the deal. Okay, so, it is injecting but not projecting. But,

7:30 The shift category is a complicated category. How do we prove that there exist injective shifts? Dr. Orders, already for modules, is quite tricky. He said, why? I will not give an explicit contraction. I will set the correctional adjunct out of which the general category can exist. And that's his famous idea, his dumb adjunct. And then, when he adds an adjunct, it's very, very general. What did he say? Now, you can create a computer using the objective of a computer. That's all. That's finished. That's what it's going to be. And the theory of cohomology, or cohomology of physics, per se, is dead. Just use the general method of cohomology. Yes. And now, of course, it was a time when Goldberg, at least, was the first technological champion. It's clear that most of the work was written by Gautam before he used this paper by Gautam and Atiyah to try to make the comparison between them. And then he was the one who produced an explicit elementary construction of the objective system and the classic form. And some people tend to remember only the construction of Gautam and say, why Gautam? In the book of Cormorant, they are able to say many interesting things, including an appendix where, for the first time, we can see the exact rules about the monads and the first creatures of my knowledge. One of the first creatures of my knowledge was the monad. Okay, so, but let's go back to the video of my talk clip. Ah, the talk clip is two lines of their own.

10:00 For topological space, for growth, locally compacted growth, topological vector space, it has a notion that it doesn't die together. That's why there are many arbitrariness and arbitrariness that can make this better. And that's what it does, and so you can use the military diets or grenades and so on. Of course, once you think seriously about it, all these constructions are reduced to one. If you have two categories, you can add the category to the main concept of all factors. One category is a modern one, and the other is a new category. Both are natural concepts that people might have learned. Both are what is a factor, and I started my discussion with people like that. It should assign the end of all them, and the end of all the three, let's say, to any And of course, even if it's easy to describe within the framework of the century, I like to refer to what is a group. Because it's not, you cannot phase it into three terms. Well, for each set G, you can't think of the totality or the cohomology. But the point is that how to define, well, for the kind of a problem, you need to go down and write the axiom, write the axiom, go ahead and write the axiom, but you have to think of both of these things, chronological, quantum, what are they doing? At which level of abstraction are we working?

12:30 It means that you are working not at the level of a psychic, but at a higher level. And at a higher level means that you are leaving mathematics into the world of meta-mathematics. And that's why, at last, we'll go back in the 50s and try to give a general description of what it means by structure-free material. We've already undertaken, in a sense, the structure theory is just an expansion of an eroded program. There is an emphasis on isomorphism, transposers to do isomorphism, monomorphism, isomorphism, groups, natural extension of the algorithm. But there was a notion of groups. That means that what is crucial is the notion of hydrography. But we wanted to have that. And I think it's Eosman who really described carefully what it is. When you define the notion of why you are, in effect, considering that the water is cool, it's time to talk about the applications of the authorities, it's a fact of the category of setting with isomorphism, maybe to the same category. To any set, you have to see the class of all possible co-constructors, and to any isomorphism, you have to send it to another one to get the bijection, the architecture, to associate the two.

15:00 If you have a jeep, bus, you want one to make it. If you have a good one, you want to transport it to make it, which is amazing. So, why do you want to be fanciful? What is abstraction? Not the abstraction of one specific set, but the general notion of the good, the general notion of the out of place, etc. You really deal with the fact. And this fact, as I said, belongs to a higher level of abstraction. That's my thought from the beginning. I don't have a lot of time to do that, the notion of moving together. The groups themselves, that's it. So, and that's why Govaki tried to make a global distinction. But since they had about, so, first of all, you, Govaki, knew that you are, not only are you a movement, but also a movement, you know, a movement. And second, he knew that we had to... To go to a higher level of mathematics, but he was very bad because he had to formulate it in terms of, in terms of metamathematical terms, which were good. And I remember, from the discussion with Obagi, whether we should include himself to a master's degree in the field of science, to this kind of category. And he wanted to make his logical standard of the world, and I was assigned the duty to write a preliminary draft. I discovered mathematical terms, mathematical terms, mathematical terms, what category should I propose? Could not be, could not be. Because the history is at this point that Bobakir, Bobakir wanted to... Because of all the founding fathers of mathematics, they were very good at that. I don't remember, but they were so they said that that's what she remembered. And Kaka was there, Sel was there, Courtney was there, and all the people did a lot of homology to it. I'm very active in that group. So everyone used it. But because we are... See, we took ourselves into very strict ideologies of science, progress in science. If we had not done that, we would have put them all over correctly. So, now, what I think, of course, are no reservations for using these frameworks in an intelligent manner.

17:30 But as I say, if you want to be consistent, you have to discuss them at the metamathematical level, or you have to admit that you are Jesus. Mathematics is part of mathematics, but it's just a higher level of abstraction and, of course, you can think of two categories here, which is another way to interpret it, but we have to absorb the level of abstraction. But as soon as we do that, we may resist the security paradoxes, like 14 categories, 4 categories, here we begin to consider the categories, etc., etc. And, right now, if you want to define the theory of quantum mechanics as a theory, you have to call it a theory. But we're going to define it as a high-level theory. And the point is that it's well known. It's well known. It's known to well serve in this type theory that the reflex, the reflex of the experiment makes logic more difficult. Because if you, if you describe the theory of this type of object as a certain level of time cycle, So, that's what they were doing. So, he went through, he went through. But, of course, you have to discover that to speak of the category of the fact that it is difficult, while, of course, you can concentrate, see this, see this, see this, see this, see this, see this, see this, see this, see this, see this, see this. But, of course, it's a realistic category. These all have a view of God, yourself, and your world. So, and you all know that before, right, that you all know that some people published debates and you proved that every factor has an adjoint by going to some 50 weeks, and so on. So, and then I've pointed out that there are more sort of analytical paradoxes connected with the so-called Nisha paradoxes.

20:00 Which is a set of defined numbers, defined numbers only when controlled and in relation to the architecture field. Okay, so Benabou pointed out that when you see, well, if I gaze you, gaze you over the space, ask the set of the different classes of shapes or vectors, it doesn't really get that easy. Because of the redacted Jewish paradox, I mean, if you use freely this notion of isomorphic tasks, of course, you have a modular and modular problem are everywhere, don't you? But, don't be tempted. So that makes me too fast forward. That's it. What is the present situation? If we have, it has to be categorized. Using mathematical methods is very easy, but you can also use small categories. Assume that every building has four categories, but you cannot do it in that complete. Or, but you can also formulate in terms of primitive notion of a primitive notion, but primitive relation, composition, etc. But the same theory is easy, and is axiomatized in an acceptable way in the Zermatt-Rohm factor, with or without choice, axiom of trichism. The two are widely defined by the Zermatt-Rohm equation, and by the standard Zermatt-Rohm equation, which Zermatt-Rohm found relatively easy. But when you put them together, they don't match, for the reason I mentioned. Of course, there is Q tried, for instance. In both the degrees there we produce the idea of universe, which was still seen in the book of De Masio and Gabriele on algebraic books of the present time.

22:30 In Nemesia, the method of a very good producer for the universe was domain in algebraic literature. He created a very, very deep field. The method was not very satisfying because of the definition. And the reflection principle forces you, once you have a universal domain, to consider it as part of a larger universal domain called the 1-2-3. So, on this issue, these universes are not very complicated. And also, they are not very pleasant. Also, they lead you to very complicated programming sessions, They are just producing arbitrary difficulties. My advice is to think of what people did. In the 18th century, people spoke freely of infinitesimal. They were never given what very clearly was. They were happy to point out the contradictions. And infinitesimal is sometimes 0, sometimes 1. I could speak about the solution of my question at least once, which I don't recommend to the pedagogy, but if you read, if you read carefully, you can see the technical point of order of much more important than this. Why don't you accept that if you do something like this, that they obey a certain rule? And that might mean that they are more or less in the same seat not to do something. What is forbidden is clear, what is forbidden is less clear. But if you accept that mathematics at a given time is not completely formal, that there is something at the borderline, borderline, and it cannot at the moment be completely formal, it's like, like, you know, for example, the level of smart, and that it's better to see the level of smart.

25:00 So, in mathematics, we have something similar. There are tetras, but why not? And the probability is still a little way from that to be unsafe. But anyway, back to the allegory, which was in Chicago in the 50s. At that time, Chicago was unsafe, maybe. Maybe more than today, I don't know. But at least even the Dalai Lama was on his head. He said, well, if you have to leave your car to go at night, better to go away, leave your car to go free, and put your car just below the big corner with the big light. You put your car in the dark corner, you take a chance. And hermaphrodites can see. So in mathematics you see. Axiomatic is a good light all over. If you move it over... It is in the axiomatic game. This is a part of mathematics because when you just do it, you find it in a very different way. If you just look at this, you are in the same area. You should go to the book and... You're saying the growth is not in the category of physicists like you have in China. Can you talk more about how we would have found that out at the time when we would have said there are no more people and at the time there were no more people in China?

27:30 It's not the opposite. I think development started from China without making sense. Yeah, when you're not using a carbon-7 carbon, it's creating carbon. It's creating a lot of carbon this time, and it's protecting you, but it's a bit of a factor. But, when there is carbon, it's fine. Well, how can we make sure that it's easy to make a carbon-7 carbon paper? Isn't there a proof? Well, you just need to be aware of that. Yeah, it should be there for all of us. They're working on it. I don't have the time to go into all of them, but people were not old enough to know about the genome. That was the question, and that was the reason I write what I call plasticity, because there is a genome which is not plastic. That means it can be different. And that's why if you give a question, it's better to have two people. One, because the plasticity of one group is not the same as the plasticity of the other group. And so on and so forth.

30:00 Well, this is why I think it's not self-evident. All of the things we should have, everybody should have. Now, I go to the U.S. and I take all the people I meet up on the field and meet them up here. The category you move on to. And you know it. And you do know a notion of it. And you express everything to them. Once you understand it, it's over. So, you know, if you go to the U.S., you're going to find that there's a revolution. Yeah, I know. There's less revolution. But now, come back to... Yeah, I know. Change of, change of. I have one proof that it works. It's that it bends. And that it bends. That's exactly it. God above takes the compares, X's and Z's, and I draw the find of them. So sheep over X, and sheep over X. That's just a collection of thoughts. So sheep is a group, and a collection of thoughts. Two groups. That's exactly it. So, I bet that reduces the question of innovation. Well, of course, it is inside what has mattered.

32:30 We are not in fact in the midst of a tri-quadrillion. To be one of us, all of us, both, can take a big conversation. Does it deserve you to go to the border? I don't give a federal question. We do in the world. No, it's clear. The idea of the border, to combine the borders of the world, before both the things are opposed and unopposed, the size is clear. Okay, something that's mystified me for a long time in being in this business is growth and acceptance. Could you comment on that? Because it seems like there's an enormous resonance, you know, between the property of the observer and climate change in all these areas about health, life, and so on. Well, I took up the game. Third of all, at the end of his day at Agnesius, I've just worked for my team. That's it. He tried to understand, ask them to make more answers. He asked them to call him. The function of mathematics was very pure, very formalistic. Mathematicians were very good at it, but very poor, and he found one way to use the notion of mathematics, which was to put one way to use it, and the second way to use it, and the third way to use it, and the fourth way to use it, and the fifth way to use it, and the sixth way to use it, and the sixth way to use it, and the seventh way to use it,

35:00 But there was always something in the life of the equation, physics implies, implies, implies, and for him, according to this point, to him, physics was the word of the day. So, here has been, so, for me, an exhibition of the world of mathematics, which is the outcome of web-watching, so, of science, of mathematics, of the world of mathematics, of the world of mathematics, of the world of mathematics, of the world of mathematics, of the world of mathematics.

37:30 At the time, I said, I also wrote a paper about this. At the end of it, I couldn't say anything. But, at some point, he was obsessed with it. Again, he said, I think, so many terms, but he said, well, look, maybe it was a point of credit. It was a good point. They don't think that the speed of life is the final part of the problem. It's about how to achieve this data. Very interesting. I'm sorry, but what about maybe five, a year? Only five, thirty? That's it. It's too big a question. You said that in a session during this day and year? Yes, yes. That's not such a big deal. And the point is that someone who is so aware of the idea of invariance should not understand what is the physical difference between the arbitrariness of activities, the dimension of analysis, quantum physics, the dimension of physics, or the dimension of activities. It is distinct what depends on the truth and the concern. I would say that the oldest of topology in my knowledge is very reliable and it is in the langs of the 9th century. It is hard to find one of those applications of mathematics that I can take in and spend so much money on. But exactly what I picked up from the differential of mathematics. Topology in two ways is something that I cannot be confident of. If I knew mathematics, I would say that it is one of the most beautiful things in the world.

40:00 And so, of course, physics have now been working when I wrote my paper in Berlin, in Pittsburgh. That is a curveball of showing that physics and quantum mechanics, Atiyah, had something to do together. Is there an anecdote that, I don't think it originated from you, but from somebody who was in contact with him, about some phone call in which... He said, you know, I have, you know, I've solved all the problems of physics and stuff like that. And I will tell you the solution if you tell me what a meter is. It's probably a very good answer. Well, I mean, you're a physicist, of course. Okay, thanks. Pierre, on this general question of the, you know, the increasing... There is increasing recognition of the centrality of the structure of functions from one category to another. How does the Tohoku paper relate to the developments going on at the same time in scheme theory? Was that slightly earlier? That was a lot earlier. So it was known already at that time in the 19th century that the category of schemes does not have a full and faithful country to the category of sets. The theory of schemes. No, no, no, no. Topical people were from Eastern Europe. Right. And it's just after that that... Oh, so the scheme development was later. Oh, right. Okay, that's what I heard. No, I think it was later. Okay, that was later. I went to work. So he was not aware at the time that he wrote a book that the, you know, that there is a political function that we've never seen in the sense that it's done. Okay. And, well, I mean, I said, I said it was a formulation. All of these have been developed as the reason I was able to get this to be a vital toolset. My thought of it was that you have to scheme us to afford it, but which was developed later on by you, which is now the way that I remember.

42:30 And I mean, I can provide you with a generalized electric system, which now comes from a substitute by the way that I remember. But it came later on. And it was at that time when you all know people, they are not children. He invested in Archival Geometry. The paper was written out in 2005 and then he came back to Paris at that time and there was a Chevalier seminar. There were two Chevaliers. One from Germany, an Archival Geometry, a foundation of Archival Geometry, and where the word scheme was invented. There were two seminars, I should say. The first one, Foundations of Mathematics, was the first one in this campus, in Nassau, and the second one was there, in Madrid. But, at that time, I was quite keen about the history of mathematics. There is a paper which is also a copy of Schoenberg's, and some back in 1949 or 1950, after the appearance of the textbook. A very big amount of foundation algebraic geometry. Then, you know, if you think of, like, geography, it's always the same restriction as a very big, irreducible variety, which, you know, a lot of variety, which may be difficult and difficult to find. Before then, people understood it's still the algebraic geometry of function. Fields, okay, fields. And that was the case with this lecture. And the reason is that there was a problem, there was a problem of tens of different theories, essentially one of which I don't know, which I don't know, which I don't know, which I don't know, which I don't know, which I don't know, which I don't know, which I don't know, which I don't know, which I don't know, which I don't know, which I don't know.

45:00 And now Chevalier took from Zeiss and his own work the idea that what is crucial in mathematics is first of all the functioning of the system, and the collection of all components associated with all components of the system. It was very important too in the decision that I made. Why? Because the basic notions will transform you. No one will point you to something higher than it. So you have to do things side by side. When you do the blow-up, I mean, there is a change in you, and that is because all the way that's really what Jesus tried to do when he was in the United States. And so, Chevrolet can't speak without the support of the missionaries. He said, an algebraic variety will have a scheme of an academic lecture. Scaffold, scheme, it's not a variety of an academic lecture, it's a scheme of an academic lecture. One of the problems is, so the function is kind of predictable. Now, to any discovered applied, applied varieties, you just have to see the function, because suddenly, it is my idea. And then... All such areas are asked to generate automatic recollections, all the local ones. And then if you want to do that, if you want to do that, you consider, right, learning how to protect the same things. And then each one generates a recollection of the same thing. This strategy was well known to the protectors, like Zaniski. But Zaniski always considered having a protective reality. But they may also be considered as, as you know, sometimes, like, for example, the Icelandic language has been promised in Agatha. And then, so, but then, Schumann expanded this to the Indian language. It's a collection of languages. That's what we're talking about here.

47:30 And then, then, well, to the food, the rice, the rice, I might use the same thing. And they should be very good at that. But it's only one patient. Very shortly after. But nevertheless, it's a very hard subject. And if you try to read the history, you'll find that the document was published on the 27th of July. The report is that it was too long, it was delayed, it was too long. But in between that. Thank you very much. Thank you for your attention. I, I, I, I, I, I, I, I, I, I, I, I, I. Okay, we'll check back at you, John, with the geometry question. Okay. Okay. If you look here, the cooling back is very poor. Thank you for your attention.

50:00 Thank you for your attention. There is a general point about geometry and arithmetic. It's a very kind of philosophical thing. Yes, this is very important. Thank you. And so, given the cross-operators, you may have a box full of them. Thank you. I guess it's a little easier to just start with the chicken and butter out there and not run with the cheese. Thank you for your attention.

52:30 Thank you for watching. Thank you for your attention. Thank you for watching. Thank you for your attention. Thank you for your attention.